What is Poisson: Definition and 507 Discussions

In probability theory and statistics, the Poisson distribution (; French pronunciation: [pwasɔ̃]), named after French mathematician Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
For instance, a call center receives an average of 180 calls per hour, 24 hours a day. The calls are independent; receiving one does not change the probability of when the next one will arrive. The number of calls received during any minute has a Poisson probability distribution: the most likely numbers are 2 and 3 but 1 and 4 are also likely and there is a small probability of it being as low as zero and a very small probability it could be 10. Another example is the number of decay events that occur from a radioactive source during a defined observation period.

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  1. Y

    Poisson equation general solution

    Homework Statement Given that \nabla2 1/r = -4\pi\delta3(r) show that the solution to the Poisson equation \nabla2\Phi = -(\rho(r)/\epsilon) can be written: \Phi(r) = (1/4\pi\epsilon) \int d3r' (\rho(r') / |r - r'|) Homework Equations The Attempt at a Solution I know...
  2. K

    Poisson counting process & order statistics

    Theorem: Let {N(t): t≥0} be a Poisson process of rate λ. Suppose we are given that for a fixed t, N(t)=n. Let Ti be the time of the ith event, i=1,2,...n. Then the (conditional) density function of Tn given that N(t)=n is the exactly the same as the density function of X(1)=min{X1,X2,...,Xn}...
  3. K

    Explaining the Joint Distribution of T1,T2,...,Tn given N(t)=n

    Let {N(t): t≥0} be a Poisson process of rate λ. We are given that for a fixed t, N(t)=n. Let Ti be the time of the ith event, i=1,2,...,n. Then the event {T1≤t1, T2≤t2,...,Tn≤tn, and N(t)=n} occurs if and only if exactly one event occurs in each of the intervals [0,t1], (t1,t2]...
  4. S

    Help with Poisson problem

    I have a Poisson-based question that I am not sure how to approach and solve. A processor receives groups of bits with a Poisson arrival rate of L. The probability of an error in receiving an erroneous bit is p. The number of bits in a group of bits is Poisson with mean M. If there is no error...
  5. T

    Poisson brackets, commutators, transformations

    Hi all, I've taken a two-course undergrad QM sequence and have been reading Shankar's Principles of Quantum Mechanics. There is some reference to the similarity between the Poisson bracket in Hamiltonian mechanics and the commutator in QM. E.g. \{x, p\} = 1 (PB) [x, p] = i \hbar...
  6. J

    Social Networks, Poisson, And ARMA

    Social Properties and First order Links I wasn't sure to put this in the math or sociology form but I already have two Social Networks topics posted in the Math forum and I think I would like to devote more specific topics to the math forum. You are subscribed to this thread Erdős–Rényi...
  7. B

    Are Poisson and Uniform Distributions Paradoxical on a Finite Line?

    Hi, all, Let's say we deploy some random points on a line of finite length according to a poisson distribution of density \lambda. Can I say that these points are also "uniformly" distributed on the same line? thks
  8. P

    How do I change a poisson spreadsheet into a bivariate version?

    I have an excel spreadsheet that uses poisson to figure out the probability of correct scores in soccer matches. How do I amend the spreadsheet to use a bivariate poisson distribution?
  9. X

    Statistics - Poisson distribution.

    (Not sure if I should have posted this in the h/w problem section since it's not really hw...just a problem I've faced recently. But if it should be there, I can move it there. ) There are 5 boxes. Each box may contain a certain amount of marbles (1, 2, 3 etc.) and some have no marbles at...
  10. F

    Poisson random variable problem

    The children in a small town own slingshots. In a recent contest 4% of them were such poor shots that they did not hit the target even once in 100 shots. If the number of times a randomly selected child has hit the target is approximately a Poisson random variable, determine the percentage of...
  11. A

    Determining Bias of MLE of k in Poisson RP

    Poisson RP: MLE of "k" P(n,tau) = [ [ (k*tau)^n ] / n! ] * exp(-k*tau) Parameter k is the process of an unknown non random variable that I want to estimate. I have determined that k^ML = [1 / (n*tau) ] sigma (xi) I believe this is correct... How do I determine if K^ML is biased?
  12. Z

    How can I solve a Dirichlet problem defined for a parabolic region?

    Homework Statement Hi, I am looking for a hint, how to solve the following Dirichlet problem. All the standard textbooks have only examples for Dirichlet problems in rectangular or polar coordinate systems, but this problem is defined for a parabolic region. Homework Equations uxx+uyy=2...
  13. U

    Transverse Displacement of Stretched String: Derivation of Poisson Eq.

    Please how would one derive the Poisson Equation model, \nabla^{2}\psi(x) = \frac{F(x)}{T}, for Transverse displacement \psi(x) of a stretched string under constant non-zero tension T and an externally applied transverse force F(x) . Assuming small angle with the horizontal (i.e...
  14. D

    Green's Function for Poisson Equation w/ Mixed BCs

    Hello I am trying to build a 3D Poisson solver using method of moments. I need to find out the Green's function for the system. My system is a rectangular box and boundary conditions are as follows: On all surfaces BC is neumann. Only on the upper and lower surface, the middle 1/3 region...
  15. T

    Fitting Poisson Distribution to Data: Need Help!

    I need to fit a Poisson distribution to this set of data (no. of counts of radioactive decay) The number of counts in a fixed time interval was recorded 500 times. With the number of counts going from 0 - 9 respectively 39 106 130 100 67 34 15 7 1 1 I understand how to use...
  16. L

    Poisson distribution and binomial distribution questions

    Please help with this thanks :) 1. (a) Define the Poisson probability distribution with mean μ. (b) Write down the binomial distribution for x successes in n independent trials each with probability p of success. (c) On average, 0.15% of the nails manufactured at a factory are known to...
  17. E

    Solving Poisson's Equation for MOSFETs: Analytic vs. Finite Difference Approach

    Hi I am working with MOSFETs and in this context I am trying to solve poissons equation inside a MOSFET. Only in the direction from the gate through the oxide and into the silicon. I know the analytic solution but now I want solve Poissons equation with the use of finite difference...
  18. V

    Poisson Distrib: Prob 10 Tubes Show Growth

    Homework Statement A source of liquid is known to contain bacteria, with the mean number of bacteria per cubic centimeter equal to 3. Ten 1 c.c. test tubes are filled with liquid. Calculate the probability that all 10 test tubes will show growth, that is contain at least 1 bacterium each. (use...
  19. R

    Calculate Poisson Bracket [H,Lz] in Cartesian Coords

    Homework Statement Calculate the Poisson bracket [H, Lz] in Cartesian Coords. Transform your result to cylndrical coords to show that [H, Lz] = -dU/dphi (note: partial derivs), where U is the potential energy. Identify the equivalent result in the Lagrangian formulation Homework Equations...
  20. A

    Simplifying the Poisson Distribution Formula: Integration and Infinite Series

    b]1. Homework Statement [/b] prove that \sum( (e^(-u)) (u(^(x)) )/x! (from x=o to n ) = \int ( (e^(-y))(y^n) )dy/n! (from u to infinite ) Homework Equations The Attempt at a Solution i know that the left is Poisson distribution formula but how to do with the 'sum' ? and...
  21. T

    Stochastic Processes - Poisson Process question

    I had this problem on my last midterm and received no credit for these parts. 1. Express trains arrive at Hiawatha station according to a Poisson process at rate 4 per hour, and independent of this, Downtown local buses arrive according to a Poisson process at rate 8 per hour. a. Given that 10...
  22. B

    Short half life nuclear homogenous or non homogenous Poisson?

    I have completed an experiment to measure the decay rate of an isotope, and I am trying to estimate uncertainties. The half life is 40 seconds, with decays counted over a 15 second period (with many of these 10 second periods for a total of 6 minutes of recordings) However in more detailed...
  23. Q

    Question about probability and poisson process

    Hi all, I have a question about probability. Can you help me? There are 2 events: - Customer A arrives the system B in accordance with a Poisson process with rate Lambda1 - Customer A arrives the system C in accordance with a Poisson process with rate Lambda2. Given that Poisson...
  24. C

    Poisson Distribution: Find E[N ∑Nᵢ₁Xᵢ]

    Homework Statement Let N,X1, X2, ... be independant random bariables where ?N has a poission Distribution with mean 3 while X1, X2... each has a poisson distribution with mean 7 Determine E[N \sum^N_{i=1} X_i] Homework Equations The Attempt at a Solution E[N \sum^N_{i=1} X_i]...
  25. S

    Solving Poisson Distribution Homework: Find f(y)

    Homework Statement In a Poisson process with intensity λ, let X1 be the time until the first event and let X2 be the time between the first and the second event. Let Y be the time until the second event, that is, Y = X1 + X2. Find the density function f(y). 2. The attempt at a solution...
  26. Q

    A problem related to Poisson process

    Hi all, I have a probability problem. Can you help me? Thank you! Here is the problem: Consider the queueing system, there are n customers 1, 2, ...N. Customer 1 arrives in accordance with a Poisson process with rate Lamda, customer 2 arrives in accordance with a Poisson process with rate...
  27. Peeter

    Fourier transform solution to electrostatics Poisson equation?

    Am just playing around, and following examples of Fourier transform solutions of the heat equation, tried the same thing for the electrostatics Poisson equation \nabla^2 \phi &= -\rho/\epsilon_0 \\ With Fourier transform pairs \begin{align*} \hat{f}(\mathbf{k}) &= \frac{1}{(\sqrt{2\pi})^3}...
  28. B

    Poisson Distribution homework

    Homework Statement An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed? [b]2. Homework Equations [/]...
  29. S

    Poisson variable w/ uni. dist. parameter

    Homework Statement Let X\sim Poi(\lambda) and assume \lambda\sim Uni(0,5) Q: Find \mathbb{P}\{X \geq 3\}Homework Equations For a Poisson r.v. with parameter lambda, \mathbb{P}\{X = k\}=\frac{\lambda^{k}e^{-\lambda}}{k!} and the probability that lambda is in the interval (0,5) is 1/5 and 0...
  30. haushofer

    Poisson brackets for Hamiltonian descriptions

    Hi, I have a (maybe rather technical) question about the Hamiltonian formulation of gauge theories, which I don't get. With an infinitesimal symmetry on your space-time M one can look at the corresponding transformation of the canonical variables in phase-space PS. This can be done by a phase...
  31. M

    Confused about Poisson equation

    Hello everybody I've been searching this today but I am a bit lost now. I've encountered two forms of Gauss law in its differential form, Poisson equation : del2V(r) = -p(r)/e del2V(r) = -4*pi*p(r)/e where V:e.potential, p:charge density, e:permivity Now, what's the difference...
  32. F

    Probability using Poisson Distribution

    Homework Statement Suppose a typographical errors committed by a typesetter occurs randomly. If that a book of 600 pages contains 600 such errors, calculate the probability by using Poisson's distribution. i) that a page contains no errors ii) that a page contains at least three errors...
  33. G

    Verifying General Solution of 2D Poisson Equation

    Hi Homework Statement Verify, that u(\vec{x}) := - \frac{1}{2 \pi} \int \limits_{\mathbb{R}^2} \log ||\vec{x} - \vec{y} || f(\vec{y}) d \vec{y} is the general solution of the 2 dimensional Poisson equation: \Delta u = - f where f \in C^2_c(\mathbb{R}^2) is...
  34. K

    Applying Poisson Equation for Electrostatic Potential in a Spherical Shell

    In many book I read, problems for electrostatic potential always lead to solving Poisson equation. I saw a problem about a spherical shell carrying some amount of charges uniformly on the surface with density \rho, and then someone put a small patch on the sphere. The patch is then made a...
  35. S

    Poisson and Helmholtz equations

    Hello guru's, I've been trying to figure out a way to incorporate an electric field source in the Helmholtz equation, and have been accumulating lots of question marks in my head. So in case of no static charge, \nabla^{2} E - \mu (\epsilon\frac{d^{2}}{dt^{2}} + \sigma\frac{d}{dt}) E = 0...
  36. D

    Evaluating Poisson Brackets: H=p^2/2m+V?

    This is a general question. When evaluating Poisson brackets, can we assume that H = p^2/2m + V?
  37. R

    Confusion on Poisson and Binomial Distribution

    Hey guys, Can anyone please explain the differences between binomial and poisson distribution. THANK U>>>>>>>>>>>>>>>>>>>>>
  38. B

    Poisson Process Homework: Chance of Mushrooms in One Yard

    Homework Statement If you find a mushroom, what is the chance that at least one more will be within one yard from it ? What is the chance that there is exactly one mushroom within the distance one yard from the point you stay? The mushrooms grow in a forest randomly , with density 0.5...
  39. T

    Poisson Distribution problem

    Homework Statement The number of customers entering a cafe during tea time is known to be poisson distribution with λ = 5. on a particular day, given that at least 2 customers have entered the cafe during the tea time. what is the probability that at least 1 more customers will enter the cafe...
  40. C

    Answer: Solve Poisson Distribution Prob | Rare Event?

    [SOLVED] Poisson Distribution Homework Statement Let X be the number of people entering the ICU in a hospital. From Historical data, we know the average number of people entering ICU on any given day is 5 a) What is the probability that the number of people entering the ICU on any given...
  41. H

    Binomial, Poisson and Normal Probability distribution help

    Binomial, Poisson and Normal Probability distribution help! Hey everyone, I've just started a new section on probability (argh) in my math course, and unlike most other maths, I cannot cope! Anyway, I was wondering if anyone could tell me if I did the following question correctly! Any helps...
  42. A

    Explore the Poisson Superfish Code

    hello, I want to know that the poisson superfish code uses finite difrerence method or finite element method?
  43. D

    Maximum likelihood of Poisson distribution

    Homework Statement Suppose X has a Poisson distribution with parameter lambda. Given a random sample of n observations, Find the MLE of lambda, and hat lambda. Find the expected value and variance of hat lambda. Show that hat lambda is a consistent estimator of lambda. Homework...
  44. K

    Laplace and Poisson Equation in oblate and prolate spheroids

    Hi everyone, I have been trying to solve both laplace and poisson equation using method of separation of variable but is giving me a hard time. Pls can anyone refer me to any textbook that solve this problem in great detail? Thanks
  45. L

    Finding the MLE of a Poisson Distribution

    Homework Statement Suppose that X has a poisson distribution with parameter \lambda . Given a random sample of n observations, find the MLE of \lambda , \hat{\lambda} . Homework Equations The MLE can be found by \Sigma^{n}_{i=1} \frac{e^{- \lambda} \lambda^{x_{i}}}{x_{i}!} = e^{-...
  46. G

    Poisson Brackets Explained: Understanding the Relationship between {x,p} = 1"

    can anyone tell me why the poisson brackets for {x,p} = 1 ..from (dx/dx)(dp/dp) - (dx/dp)(dp/dx)... shouldn this equal 0??
  47. T

    Green identity, poisson equation.

    Suppose \phi is a scalar function: R^n\to R, and it satisfies the Poisson equation: \nabla^2 \phi=-\dfrac{\rho}{\varepsilon_0} Now I want to calculate the following integral: \int \phi \nabla^2 \phi \,dV So using Greens first identity I get: \int \phi \nabla^2 \phi \,dV = \oint_S \phi...
  48. B

    Calculating Probabilities for Poisson Distribution

    Homework Statement Phone calls are received at Diane residence have a Poisson distribution with \lambda =2. a) If Diane takes a shower for 10 min, what is the probability that the phone rings Once or Twice. b) How long can she shower if the probability of receiving no calls be at most 0.5...
  49. W

    Can the Axisymmetric Poisson Equation for Magnetostatics be Solved?

    For a magnetostatics problem I seek the solution to the following equation \frac{1}{x}\frac{d}{dx} \left( x \frac{dy(x)}{dx} \right) = -C^2 y(x) (C a real constant) or equivalently x \frac{d^2 y(x)}{dx^2} + \frac{dy(x)}{dx} + C^2 x y(x)=0 It seems so simple, but finding a...
  50. M

    Bivariate poisson - probability

    Let {Mi(t), t >= 0 }, i=1, 2 be independent Poisson processes with respective rates λi, i=1, 2, and set N1(t) = M1(t) + M2(t), N2(t) = M2(t) + M3(t) The stochastic process {(N1(t), N2(t)), t >= 0} is called a bivariate Poisson process. (a) Find P{N1(t) = n, N2(t) = m} (b) Find Cov...
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